open problems in proof-theoretic semantics · nature of hypotheses and the problem of the...

Open Problems in Proof-TheoreticSemantics

Peter Schroeder-Heister

Abstract I present three open problems the discussion and solution of which Iconsider relevant for the further development of proof-theoretic semantics: (1) Thenature of hypotheses and the problem of the appropriate format of proofs, (2) theproblem of a satisfactory notion of proof-theoretic harmony, and (3) the problem ofextending methods of proof-theoretic semantics beyond logic.

Keywords Proof-theoretic semantics · Hypothesis · Natural deduction · Sequentcalculus · Harmony · Identity of proofs · Definitional reflection

1 Introduction

Proof-theoretic semantics is the attempt to give semantical definitions in terms ofproofs. Its main rival is truth-theoretic semantics, or, more generally, semantics thattreats the denotational function of syntactic entities as primary. However, since thedistinction between truth-theoretic and proof-theoretic approaches is not as clear cutas it appears at first glance, particularly if ‘truth-theoretic’ is understood in its model-theoretic setting (see Hodges [33], and Došen [10]). it may be preferable to redirectattention from the negative characterisation of proof-theoretic semantics to its posi-tive delineation as the explication ofmeaning throughproofs. Thus,we leave aside thequestion of whether alternative approaches can or do in fact deal with the phenomenathat proof-theoretic semantics tries to explain. In proof-theoretic semantics, proofsare not understood simply as formal derivations, but as entities expressing argumentsby means of which we can acquire knowledge. In this sense, proof-theoretic seman-tics is closely connected and strongly overlaps with what Prawitz has called generalproof theory.

The task of this paper is not to provide a philosophical discussion of the valueand purpose of proof-theoretic semantics. For that the reader may consult Schroeder-Heister [56, 61] and Wansing [72]. The discussion that follows presupposes some

P. Schroeder-Heister (B)Department of Computer Science, University of Tübingen, Tübingen, Germanye-mail: [email protected]

© The Author(s) 2016T. Piecha and P. Schroeder-Heister (eds.), Advances in Proof-Theoretic Semantics,Trends in Logic 43, DOI 10.1007/978-3-319-22686-6_16

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254 P. Schroeder-Heister

acquaintance with basic issues of proof-theoretic semantics. Three problems areaddressed, which I believe are crucial for the further development of the proof-theoretic approach. This selection is certainly personal, and many other problemsmight be added. However, it is my view that grappling with these three problemsopens up further avenues of enquiry that are needed if proof-theoretic semantics isto mature as a discipline.

The first problem is the understanding of hypotheses and the format of proofs. It isdeeply philosophical and deals with the fundamental concepts of reasoning, but hasimportant technical implications when it comes to formalizing the notion of proof.The second problem is the proper understanding of proof-theoretic harmony. Thisis one of the key concepts within proof-theoretic semantics. Here we claim that anintensional notion of harmony should be developed. The third problem is the needto widen our perspective from logical to extra-logical issues. This problem proceedsfrom the insight that the traditional preoccupation of proof-theoretic semantics withlogical constants is far too limited.

I work within a conventional proof-theoretic framework where natural deductionand sequent calculus are the fundamental formal models of reasoning. Using catego-rial logic, which can be viewed as abstract proof theory, many new perspectives onthese three problems would become possible. This task lies beyond the scope of whatcan be achieved here. Nevertheless, I should mention that the proper recognition ofcategorial logic within proof-theoretic semantics is still a desideratum. For the topicof categorial proof theory the reader is referred to Došen’s work, in particular to hisprogrammatic statement of 1995 [6], his contribution to this volume [11] and thedetailed expositions in two monographs [7, 12].

2 The Nature of Hypotheses and the Format of Proofs

The notion of proof from hypotheses—hypothetical proof—lies at the heart of proof-theoretic semantics. A hypothetical proof is what justifies a hypothetical judgement,which is formulated as an implication.However, it is not clearwhat is to be understoodby a hypothetical proof. In fact, there are various competing conceptions, often notmade explicit, which must be addressed in order to describe this crucial concept.

2.1 Open Proofs and the Placeholder View

The most widespread view in modern proof-theoretic semantics is what I have calledthe primacy of the categorical over the hypothetical [58, 60]. According to thisview, there is a primitive notion of proof, which is that of an assumption-free proofof an assertion. Such proofs are called closed proofs. A closed proof proves outright,without referring to any assumptions, that which is being claimed. A proof fromassumptions is then considered an open proof, that is, a proof which, using Frege’s

Open Problems in Proof-Theoretic Semantics 255

term, may be described as ‘unsaturated’. The open assumptions are marks of theplaces where the proof is unsaturated. An open proof can be closed by substitutingclosed proofs for the open assumptions, yielding a closed proof of the final assertion.In this sense, the open assumptions of an open proof are placeholders for closedproofs. Therefore, one can speak of the placeholder view of assumptions.

Prawitz [46], for example, speaks explicitly of open proofs as codifying openarguments. Such arguments are, so-to-speak, arguments with holes that can be filledwith closed arguments, and similarly for open judgements and open grounds [50].A formal counterpart of this conception is the Curry-Howard correspondence, inwhich open assumptions are represented by free term variables, corresponding to thefunction of variables to indicate open places. Thus, one is indeed justified in viewingthis conception as extending to the realmof proofs Frege’s idea of the unsaturatednessof concepts and functions. That hypotheses are merely placeholders entails that nospecific speech act is associated with them. Hypotheses are not posed or claimed butplay a subsidiary role in a superordinated claim of which the hypothesis marks anopen place.

I have called this placeholder view of assumptions and the transmission view ofhypothetical proofs a dogma of standard semantics. It should be considered a dogma,as it is widely accepted without proper discussion, despite alternative conceptionsbeing readily available. It belongs to standard semantics, as it underlies not only thedominant conception of proof-theoretic semantics, but in some sense it also underliesclassical truth-condition semantics. In the classical concept of consequence accordingto Bolzano and Tarski, the claim that B follows from A is justified by the fact that,in any model of A—in any world, in which A is true—, B is true as well. Thismeans that hypothetical consequence is justified by reference to the transmission ofthe categorical concept of truth from the condition to the consequent. We are notreferring here to functions or any sort of process that takes us from A to B, butjust to the metalinguistic universal implication that, whenever A is true, B is true aswell. However, as with the standard semantics of proofs, we retain the idea that thecategorical concept precedes the hypothetical concept, and the latter is justified byreference to the former concept.

The concept of open proofs in proof-theoretic semantics employs not only the ideathat they can be closed by substituting something into the open places, but also theidea that they can be closed outright by a specific operation of assumption discharge.This is what happens with the application of implication introduction as describedby Jaskowski [34] and Gentzen [25]. Here the open place disappears because whatis originally expressed by the open assumption now becomes the condition of animplication. This can be described as a two-layer system. In addition to hypotheticaljudgements given by an open proof with the hypotheses as open places, we havehypothetical judgements in the form of implications, in which the hypothesis is asubsentence. A hypothetical judgement in the sense of an implication is justified byan open proof of the consequent from the condition of the implication. The idea oftwo layers of hypotheses is typical for assumption-based calculi and in particular fornatural deduction, which is the main deductive model of proof-theoretic semantics.


According to the placeholder view of assumptions there are two operations thatone can perform as far as assumptions are concerned: Introducing an assumptionas an open place, and eliminating or closing it. The closing of an assumption canbe achieved in either of two ways: by substituting another proof for it, or by dis-charge. In the substitution operation the proof substituted for the assumption neednot necessarily be closed. However, when it is open then instead of the original openassumption the open assumptions of the substitute become open assumptions of thewhole proof. Thus a full closure of an open assumption by means of substitutionrequires a closed proof as a substitute. To summarise, the three basic operations onassumptions are: assumption introduction, assumption substitution, and assumptiondischarge.

These three basic operations on assumptions are unspecific in the following sense:They do not depend on the internal form of the assumption, that is, they do notdepend on its logical or non-logical composition. They take the assumption as it is.In this sense these operations are structural. The operation of assumption discharge,although pertaining to the structure of the proof, is normally used in the contextof a logical inference, namely the introduction of implication. The crucial idea ofimplication introduction, as first described by Jaskowski and Gentzen, involves that,in the context of a logical inference, the structure of the proof is changed. Thisstructure-change is a non-local effect of the logical rule.

The unspecific character of the operations on assumptions means that, in theplaceholder view, assumptions are notmanipulated in any sense. The rules that governthe internal structure of propositions are always rules that concern the assertions, butnot the assumptions made. Consequently, the placeholder view is assertion-centredas far as content is concerned. In an inference step we pass from assertions alreadymade to another assertion. At such a step the structure of the proof, in particularwhichassumptions are open,may be changed, but not the internal form of assumptions. Thismay be described as the forward-directedness of proofs. When proving something,we may perform structural changes in the proof that lies ‘behind’ us, but withoutchanging the content of what lies behind. It is important to note that we do not herecriticise this view of proofs. We merely highlight what one must commit oneself toin affirming this view.

2.2 The No-Assumptions View

The most radical alternative to the placeholder view of assumptions is the claim thatthere are no assumptions at all. This view is much older than the placeholder viewand was strongly advocated by Frege (see for example Frege [22]). Frege arguesthat the aim of deduction is to establish truth, and, in order to achieve that goal,deductions proceed from true assertions to true assertions. They start with assertionsthat are evident or for other reasons true. This view of deduction can be traced back toBolzano’s Wissenschaftslehre [1] and its notion of ‘Abfolge’. This notion means the


relationship between true propositions A and B, which obtains if B holds becauseA holds.

However, in view of the fact that hypothetical claims abound in everyday life,science, legal reasoning etc., it is not very productive simply to deny the idea ofhypotheses. Frege was of course aware that ‘if …, then …’ statements play a centralrole wherever we apply logic. His logical notation (the Begriffsschrift) uses implica-tion as one of the primitive connectives. The fact that he can still oppose the idea ofreasoning from assumptions is that he denies a two-layer concept of hypotheses. Aswe have the connective of implication at hand, there is no need for a second kind ofhypothetical entity that consists of a hypothetical proof from assumptions. Insteadof maintaining that B can be proved from the hypothesis A, we should just be able toprove A implies B non-hypothetically, which has the same effect. There is no needto consider a second structural layer at which hypotheses reside.

For Frege this means, of course, that implication is not justified by some sortof introduction rule, which was a much later invention of Jaskowski and Gentzen.The laws of implication are justified by truth-theoretic considerations and codifiedby certain axioms. That A implies itself, is, for example, one of these axioms (inFrege’s Grundgesetze, [21]), from which a proof can start, as it is true.

The philosophical burden here lies in the justification of the primitive axioms.If, like Frege, we have a truth-theoretic semantics at hand, this is no fundamentalproblem, as Frege demonstrates in detail by a truth-valuational procedure. It becomesa problemwhen themeaning of implication is to be explained in terms of proofs. Thisis actually where the need for the second structural layer arises. It was the ingeniousidea of Jaskowski andGentzen to devise a two-layermethod that reduces themeaningof implication to something categorically different. Even though this interpretationwas not, or was only partly, intended as a meaning theory for implication (in Gentzen[25] as an explication of actual reasoning in mathematics, in Jaskowski [34] as anexplication of suppositional reasoning) it has become crucial in that respect in laterproof-theoretic semantics. The single-layer alternative of starting from axioms inreasoning presupposes an external semantics that is not framed in terms of proofs.

Such an external semantics need not be a classical truth-condition semantics,or an intuitionistic Kripke-style semantics. It could, for example, be a constructiveBHK-style semantics, perhaps along the lines of Goodman-Kreisel or a variant ofrealisability (see, for example, Dean and Kurokawa [4]). We would then have ajustification of a formal system by means of a soundness proof. As soon as theaxioms or rules of our system are sound with respect to this external semantics, theyare justified. In proof-theoretic semantics, understood in the strict sense of the term,such an external semantics is not available. This means that a single-layer concept ofimplication based only on axioms and rules, but without assumptions, is not a viableoption.1

1Digression on Frege: Even though formally Frege has only a single-layer system, there is a hiddentwo-layer system that lies in the background. Frege makes an additional distinction between uppermember and lower members of (normally iterated) implications. This distinction is not a syntacticalproperty of the implication itself, but something that we attach to it, and that we can attach to it


2.3 Bidirectionality

The transmission view of consequence is incorporated in natural deduction in that wehave the operations of assumption introduction, assumption-closure by substitutionand assumption-closure by discharge. There are various notations for it. The mostcommon such notation in proof-theoretic semantics is Gentzen’s [25] tree nota-tion that was adopted and popularised by Prawitz [44]. Alternative notations areJaskowski’s [34] box notation, which is the origin of Fitch’s [18] later notation. Afurther notation is sequent-style natural deduction. In this notation proofs consist ofjudgements of the form A1, . . . , An � B, where the A’s represent the assumptionsand B the conclusion of what is claimed. This can be framed either in tree form,in boxed form, or in a mixture of both. With sequent-style natural deduction theoperation of assumption discharge is no longer non-local. In the case of implicationintroduction

A1, . . . , An, B � C

A1, . . . , An � B → C(1)

we pass from one sequent to another without changing the structure of the proof. Infact, in such a proof there are no sequents as assumptions, but every top sequent isan axiom, normally of the form A � A or A1, . . . , An, A � A. However, this is nota no-assumptions system in the sense of Sect. 2.2: It combines assumption-freenesswith a two-layer approach. The structural layer is the layer of sequents composed outof lists of formulas building up the left and right sides (antecedent and succedent)of a sequent, whereas the logical layer concerns the internal structure of formulas.By using introduction rules such as (1), the logical layer is characterised in termsof the structural layer. In this sense sequent-style natural deduction is a variant of‘standard’ natural deduction.

However, a different picture emerges if we look at the sequent calculus LK orLJ that Gentzen devised. These are two-layer systems in which we can manipulatenot only what is on the right hand side and what corresponds to assertions, but alsowhat is on the left hand side and what corresponds to assumptions. In sequent-stylenatural deduction a sequent

A1, . . . , An � B (2)

just means that we have a proof of B with open assumptions A1, . . . , An :

A1, . . . , An

DB

(3)

(Footnote 1 continued)in different ways. This distinction is analogous to that between assumptions and assertion, so that,when we prove an implication, we can at the same time regard this proof as a proof of the uppermember of this implication from its lower members taken as assumptions. In the GrundgesetzeFrege [21] even specifies rules of proofs in terms of this second-layer distinction. This means thathe himself goes beyond his own idea of a single-layer system. See Schroeder-Heister [64].


In the symmetric sequent calculus, which is the original form of the sequent calculus,the A1, . . . , An can still be interpreted as assumptions in a derivation of B, but nolonger as open assumptions in the sense of the transmission view. Or at least they canbe given an alternative interpretation that leads to a different concept of reasoning.In the sequent calculus we have introduction rules not only on the right hand side,but also on the left hand side, for example, in the case of conjunction:

A, A1, . . . , An � C(∧L)

A ∧ B, A1, . . . , An � C

This can be interpreted as a novel model of reasoning, which is different fromassertion-centred forward reasoning in natural deduction. When reading a sequent(2) in the sense of (3), the step of (∧L) corresponds to:

A ∧ BADC

This is a step that continues a given derivation D of C from A upwards to a deriva-tion of C from A ∧ B. Therefore, from a philosophical point of view, the sequentcalculus presents a model of bidirectional reasoning, that is, of reasoning that, bymeans of right-introduction rules, extends a proof downwards, and, by means ofleft-introduction rules, extends a proof upwards.

This is, of course, a philosophical interpretation of the sequent calculus, reading itas describing a certainway of constructing a proof fromhypotheses.2 Since under thisschemaboth assumptions and assertions are liable to applicationof rules, assumptionsare no longer understood simply as placeholders for closed proofs. Both assumptionsand assertions are now entities in their own right. Read in that way we have a novelpicture of the nature of hypotheses. We can give this reading a format in natural-deduction style, namely by formulating the rule (∧L) as

A ∧ B[A]C

C

where the line above A ∧ B expresses that A ∧ B stands here as an assumption. Wemay call this system a natural-deduction sequent calculus, that is, a sequent calculusin natural-deduction style. It is a system in which major premisses of eliminationrules occur only as assumptions (‘stand proud’ in Tennant’s [70] terminology). The

2Gentzen [25] himself devised the sequent calculus as a technical device to prove his Hauptsatzafter giving a philosophical motivation of the calculus of natural deduction. He wanted to give acalculus in ‘logistic’ style, by which he meant a calculus without assumptions that just moves fromclaim to claim and whose rules are local due to the assumption-freeness of the system. The term‘logistic’ comes from the designation of modern symbolic as opposed to traditional logic in the1920s (see Carnap, [3]).


intuitive idea of this step is that we can introduce an assumption in the course of aderivation. If a proof of C from A is given, we can, by introducing the assumptionA∧B and discharging the given assumption A, pass over toC . That A∧B occurs onlyas an assumption and cannot be a conclusion of any other rule, demonstrates that wehave a different model of reasoning, in which assumptions are not just placeholders

for other proofs, but stand for themselves. The fact that, given a proof DA ∧ B

of

A ∧ B and a proof of the form

A ∧ B

[A]D ′C

C

we obtain a proof of C by combining these two proofs is no longer built into thesystem and its semantics, but something that must be proved in the form of a cutelimination theorem. According to this philosophical re-interpretation of the sequentcalculus, assumptions and assertions now resemble handles at the top and at thebottom of a proof, respectively. It is no longer the case that one side is a placeholderwhereas the other side represents proper propositions.

This interpretation also means that the sequent calculus is not just a meta-calculusfor natural deduction, as Prawitz ([44], Appendix A) suggests. It is a meta-calculus inthe sense thatwhenever there is a natural-deduction derivation of B from A1, . . . , An ,there is a sequent-calculus derivation of the sequent A1, . . . , An � B, and vice versa.However, this does not apply to proofs. There is no rule application in natural deduc-tion that corresponds to an application of a left-introduction rule in the sequent calcu-lus. This means that at the level of proof construction there is no one-one correspon-dence, but something genuinely original in the sequent calculus. This is evidencedby the fact that the translation between sequent calculus and natural deduction isnon-trivial. Prawitz is certainly right that a sequent-calculus proof can be viewed asgiving instructions about how to construct a corresponding natural-deduction deriva-tion. However, we would like to emphasise that it can be interpreted to be more thansuch a metalinguistic tool, namely as representing a way of reasoning in its ownright. We do not want to argue here in favour of either of these positions. However,we would like to emphasise that the philosophical significance of the bidirectionalapproach has not been properly explored (see also Schroeder-Heister, [59]).

2.4 Local and Global Proof-Theoretic Semantics

We have discussed the philosophical background of three conceptions of hypothesesand hypothetical proofs. Each of them has strong implications for the form of proof-theoretic semantics. According to the no-assumptions view (Sect. 2.2) with its single-layer conception there is no structural way of dealing with hypotheses. Therefore,


there is no proof-theoretic semantics of implication, at least not along the commonline that an implication expresses that we have or can generate a hypothetical proof.We would need instead a semantics from outside.

A proof-theoretic semantics for the placeholder view of assumptions (Sect. 2.1),even though it is assertion-centred, is not necessarily verificationist in the sense thatit considers introduction rules for logical operators to be constitutive of meaning.Nothing prevents us from considering elimination rules as primitive meaning-givingrules and justifying introduction rules from them (seePrawitz [49], Schroeder-Heister[67]). However, the placeholder-view forces one particular feature that might beseen as problematic from certain points of view, namely the global character of thesemantics.

According to the placeholder-view of assumptions, an open derivation D of Bfrom A

ADB

would be considered valid if for every closed derivation of A

D ′A

the derivationD ′ADB

obtained by substituting the derivation D ′ of A for the open assumption A is valid.This makes sense only if proof-theoretic validity is defined for whole proofs ratherthan for single rules, since the entity in which the assumption A is an open place is aproof. A proof would not be considered valid because it is composed of valid rules,but conversely, a rule would be considered valid if it is a limiting case of a proof,namely a one-step proof. This is actually how the definitions of validity in the spiritof Prawitz’s work proceed (Prawitz [48], Schroeder-Heister [56, 61]).

This global characteristic of validity has strong implications. We must expectnow that a proof as a whole is well-behaved in a certain sense, for example, thatit has certain features related to normalisability. In all definitions of validity wehave as a fundamental property that a closed proof is valid iff it reduces to a validclosed proof, which means that validity is always considered modulo reduction. Andreduction applies to the proof as a whole, which means that it is a global issue. Asvalidity is global, there is no way for partial meaning in any sense. A proof can bevalid, and it can be invalid. However, there is no possibility of the proof being onlypartially valid as reflected in the way the proof behaves.


This global proof semantics has its merits as long as one considers only cases suchas the standard logical constants, where everything is well-founded and we can buildvalid sentences from the bottom up. However, it reaches its limits of applicability,if proof-theoretic semantics should cover situations where we do not have such afull specification of meaning. When dealing with iterated inductive definitions, wecan, of course, require that definitions be well-behaved, as Martin-Löf [40] did inhis theory. However, when it comes to partial inductive definitions, the situation isdifferent (see Sect. 4).

Here it is much easier to say: We have locally valid rules, but the compositionof such rules is not necessarily a globally valid derivation. In a rule-based approachwe can make the composition of rules and its behaviour a problem, whereas onthe transmission view the validity of composition is always enforced. Substitutionbecomes an explicit step,which canbeproblematised. Itwill be possible, in particular,to distinguish between the validity of rules and the effect the composition of ruleshas on a proof. In fact, it is not even mandatory to allow from the very beginningthat each composition of (locally) valid rules renders a proof valid. Wemight imposefurther restriction on the composition of valid rules. This occurs especially when thecomposition of locally valid rules does not yield a proof the assumptions of whichcan be interpreted as placeholders, that is, for which the substitution property doesnot hold.

Therefore, the bidirectional model of proof allows for a local proof-theoreticsemantics. Here we can talk simply of rules that extend a proof on the assertion oron the assumption side. There will be rules for each side, and one may discuss issuessuch as when these rules are in harmony or not. Whether one side is to be consideredprimary, and, if yes, which one, does not affect the model of reasoning as such. Inany case a derivation would be called valid if it consists of the application of validrules, which is exactly what local proof-theoretic semantics requires.

3 The Problem of Harmony

In the proof-theoretic semantics of logical constants, harmony is a, or perhaps the,crucial concept. If we work in a natural-deduction framework, harmony is a propertythat introduction and elimination rules for a logical constant are expected to satisfywith respect to each other in order to be appropriate. Harmony guarantees that wedo not gain anything when applying an introduction rule followed by an eliminationrule, but also, conversely, that from the result of applying elimination rules we can,by applying introduction rules, recover what we started with. The notion of harmonyor ‘consonance’ was introduced by Dummett ([13], pp. 396–397).3

3At least in his more logic-oriented writings, Dummett tends to use ‘harmony’ as comprising onlythe ‘no-gain’ direction of introductions followed by eliminations, and not the ‘recovery’ directionof eliminations followed by introductions, which he calls ‘stability’. See Dummett [14].


However, it is not absolutely clear how to define harmony. Various competingunderstandings are to be found in the literature. We identify a particular path thathas not yet been explored, and we call this path ‘intensional’ or ‘strong’ harmony.The need to consider such a notion on the background of the discussion, initiated byPrawitz [45], on the identity of proofs, in particular in the context of Kosta Došen’swork (see Došen [6, 8, 9], Došen and Petric [12]), was raised by Luca Tranchini.4

As the background to this issue we first present two conceptions of harmony, whichare not reliant on the notion of identity of proofs.

3.1 Harmony Based on Generalised Rules

According toGentzen “the introductions represent so-to-speak the ‘definitions’ of thecorresponding signs” whereas the eliminations are “consequences” thereof, whichshould be demonstrated to be “unique functions of the introduction inferences on thebasis of certain requirements” (Gentzen [25], p. 189). If we take this as our charac-terisation of harmony, we must specify a function F which generates eliminationrules from given introduction rules. If elimination rules are generated according tothis function, then introduction and elimination rules are in harmony with each other.

There have been various proposals to formulate elimination rules in a uniformwaywith respect to given introduction rules, in particular those by von Kutschera [36],Prawitz [47] and Schroeder-Heister [53]. At least implicitly, they all intend to capturethe notion of harmony. Read [51, 52] has proposed to speak of ‘general-eliminationharmony’. Formulated as a principle, we could say: Given a set cI of introductionrules for a logical constant c, the set of elimination rules harmonious with cI is theset of rules generated byF , namelyF (cI ). In other words, cI andF (cI ) are bydefinition in harmony with each other. If alternative elimination rules cE are givenfor c, one would say that cE is in harmony with cI , if cE is equivalent to F (cI )

in the presence of cI . This means that, in the system based on cI and cE , we canderive the rules contained in F (cI ), and in the system based on cI and F (cI ),we can derive the rules contained in cE .

Consequently the generalised elimination rules F (cI ) are canonical harmo-nious elimination rules5 given introduction rules cI . The approaches mentionedabove develop arguments that justify this distinguishing characteristic, for exampleby referring to an inversion principle. The canonical elimination rule ensures that

4This topic will be further pursued by Tranchini and the author.5Of course, this usage of the term ‘canonical’ is different from its usage in connectionwithmeaning-giving introduction rules, for example, for derivations using an introduction rule in the last step.


everything that can be obtained from the premisses of each introduction rule can beobtained from their conclusion. For example, if the introduction rules for ϕ have theform

�1(ϕ I) ϕ

. . .�mϕ

the canonical elimination rule takes the form

ϕ[�1]

r . . .

[�m]r

(ϕ E)can r

The exact specification of what the �i can mean, and what it means to use the �i asdischargeable assumptions, depends on the framework used (see Schroeder-Heister[63, 65]).6

While the standard approaches use introduction rules as their starting point, it ispossible in principle, and in fact not difficult, to develop a corresponding approachbased on elimination rules. Given a set of elimination rules cE of a connective c, wewould define a function G that associates with cE a set of introduction rules G (cE )

as the set of introduction rules harmonious to cI . While the rules in G (cE ) arethe canonical harmonious introduction rules, any other set cI of introduction rulesfor c would be in harmony with cE if cI is equivalent to G (cE ) in the presenceof cE . This means that, in the system based on cE and cI , we can derive the rulescontained in G (cE ), and in the system based on cE and G (cE ), we can derive therules in cI . For example, if the elimination rules have the form

ϕ �1(ϕ E) q1

. . .ϕ �m

qm

then the canonical introduction rule takes the form

[�1]q1 . . .

[�m]qm

(ϕ I)can ϕ

Here the conclusions of the elimination rules become premisses of the canonicalintroduction rules. Again, the exact specification of �i depends on the frameworkused. For example, if, for the four-place connective ∧→, the set ∧→ E consists ofthe three elimination rules

∧→ (p1, p2, p3, p4)(∧→ E) p1

∧→ (p1, p2, p3, p4)p2

∧→ (p1, p2, p3, p4) p3p4

6In this section, we do not distinguish between schematic letters for formulas and propositionalvariables, as we are also considering propositional quantification. Therefore, in a rule schema suchas (ϕ E)can, the propositional variable r is used as a schematic letter.


we would define G (∧→ E ) as consisting of the single introduction rule

p1 p2[p3]p4

(∧→ I)can ∧→ (p1, p2, p3, p4)

In Schroeder-Heister [63] functions F and G are defined in detail.

3.2 Harmony Based on Equivalence

Approaches based on generalised eliminations or generalised introductions maintainthat these generalised rules have a distinguished status, so that harmony can bedefined with respect to them. An alternative way would be to explain what it meansthat given introductions cI and given eliminations cE are in harmony with eachother, independent of any syntactical function that generates cE from cI or viceversa. This way of proceeding has the advantage that rule sets cI and cE can besaid to be in harmony without starting either from the introductions or from theeliminations as primary meaning-giving rules. That for certain syntactical functionsF and G the rule sets cI andF (cI ), or cE and G (cE ), are in harmony, is then aspecial result and not the definiens of harmony. The canonical functions generatingharmonious rules operate on sets of introduction and elimination rules for whichharmony is already defined independently. This symmetry of the notion of harmonyfollows naturally from an intuitive understanding of the concept.

Such an approach is described for propositional logic in Schroeder-Heister [66].Its idea is to translate the meaning of a connective c according to given introductionrules cI into a formula cI of second-order intuitionistic propositional logic IPC2,and its meaning according to given elimination rules cE into an IPC2-formula cE .Introductions and eliminations are then said to be in harmony with each other, if cI

and cE are equivalent (in IPC2). The introduction and elimination meanings cI andcE can be read off the proposed introduction and elimination rules. For example,consider the connective && with the introduction and elimination rules

p1[p1]p2

p1 && p2

p1&&p2[p1]r1

[p2]r2

[r1, r2]r

r

Its introduction meaning is p1 ∧ (p1 → p2), and its elimination meaning is∀r1r2r(((p1 → r1) ∧ (p2 → r2) ∧ ((r1 ∧ r2) → r)) → r). As these formulasare equivalent in IPC2, the introduction and elimination rules for && are in harmonywith each other. Further examples are discussed in Schroeder-Heister [66].

The translation into IPC2 presupposes, of course, that the connectives inherent inIPC2 are already taken for granted. Therefore, this approach works properly only forgeneralised connectives different from the standard ones. As it reduces semantical


content to what can be expressed by formulas of IPC2, it was called a ‘reductive’rather than ‘foundational’ approach. As described in Schroeder-Heister [63] thiscan be carried over to a framework that employs higher-level rules, making thereference to IPC2 redundant. However, as the handling of quantified rules in thisframework corresponds to what can be carried out in IPC2 for implications, this isnot a presupposition-free approach either. The viability of both approaches hingeson the notion of equivalence, that is, the idea that meanings expressed by equivalentpropositions (or rules in the foundational approach), one representing the contentof introduction-premisses and the other one representing the content of elimination-conclusions, is sufficient to describe harmony.

3.3 The Need for an Intensional Notion of Harmony

Even though the notion of harmony based on the equivalence of cI and cE in IPC2 orin the calculus of quantified higher-level rules is highly plausible, a stronger notioncan be considered.7 Let us illustrate this by an example: Suppose we have the set∧I consisting of the standard conjunction introduction

p1 p2p1 ∧ p2

and two alternative sets of elimination rules:∧E consisting of the standard projectionrules

p1 ∧ p2p1

p1 ∧ p2p2

and ∧E ′ consisting of the alternative rules

p1 ∧ p2p1

p1 ∧ p2 p1p2

It is obvious that ∧E and ∧E ′ are equivalent to each other, and also equivalent to therule

p1 ∧ p2[p1, p2]

qq

which is the canonical generalised elimination rule for ∧. However, do ∧E and∧E ′ mean the same in every possible sense? According to ∧E , conjunction justexpresses pairing, that is, a proof of p1 ∧ p2 is a pair 〈�1,�2〉 of proofs, one forp1 and one for p2. According to ∧E ′, conjunction expresses something different.A proof of p1 ∧ p2 is now a pair that consists of a proof of p1, and a proof of p2which is conditional on p1. Using a functional interpretation of conditional proofs,

7The content of this subsection uses ideas presented by Kosta Došen in personal discussion.


this second component can be read as a procedure f that transforms a proof of p1into a proof of p2 so that, according to ∧E ′, conjunction expresses the pair 〈�1, f 〉.Now 〈�1,�2〉 and 〈�1, f 〉 are different. From 〈�1, f 〉 we can certainly constructthe pair 〈�1, f (�1)〉, which is of the desired kind. From the pair 〈�1,�2〉 we cancertainly construct a pair 〈�1, f ′〉, where f ′ is the constant function that maps anyproof of p1 to�2. However, if we combine these two constructions, we do not obtainwhat we started with, since we started with an arbitrary function and we end up witha constant function. This is an intuition that is made precise by the consideration thatp1 ∧ p2, where conjunction here has the standard rules ∧I and ∧E , is equivalentto p1 ∧ (p1 → p2), but is not isomorphic to it (see Došen [6, 9]). Correspondingly,only ∧E , but not ∧E ′ is in harmony with ∧I .

Unlike the notion of equivalence, which only requires a notion of proof in asystem, the notion of isomorphism requires a notion of identity of proofs. Thisis normally achieved by a notion of reduction between proofs, such that proofsthat are linked by a chain of reductions are considered identical.8 In intuitionisticnatural deduction these are the reductions reducing maximum formulas (in the caseof implication this corresponds to β-reduction), as well as the contractions of anelimination immediately followed by an introduction (in the case of implication thiscorresponds to η-reduction) and the permutative reductions in the case of disjunctionand existential quantification. Using these reductions, moving from p1 ∧ p2 to p1 ∧(p1 → p2) and back to p1 ∧ p2 reduces to the identity proof p1 ∧ p2 (i.e., theformula p1 ∧ p2 conceived as a proof from itself), whereas conversely, moving fromp1 ∧ (p1 → p2) to p1 ∧ p2 and back to p1 ∧ (p1 → p2) does not reduce to theidentity proof p1 ∧ (p1 → p2). In this sense ∧E and ∧E ′ cannot be identified.

More precisely, given a formal system together with a notion of identity of proofs,two formulasψ1 andψ2 are called isomorphic if there are proofs ofψ2 fromψ1 and ofψ1 fromψ2, such that each of the combination of these proofs (yielding a proof ofψ1from ψ1 and ψ2 from ψ2) reduces to the trivial identity proof ψ1 or ψ2, respectively.As this notion,which is bestmade fully precise in categorial terminology, requires notonly a notion of proof but also a notion of identity between proofs, it is an intensionalnotion, distinguishing between possibly different ways of proving something. Theintroduction of this notion into the debate on harmony calls for a more finegrainedanalysis. We may now distinguish between purely extensional harmony, which isjust based on equivalence and which may be explicated in the ways described in theprevious two subsections, and intensional harmony, which requires additional meanson the proof-theoretic side based on the way harmonious proof conditions can betransformed into each other.

However, even though the notion of an isomorphism has a clear meaning in aformal system given a notion of identity of proofs, it is not so clear how to use it todefine a notion of intensional harmony. The notion of intensional harmony will alsobe called strong harmony in contradistinction to extensional harmony which is alsocalled weak harmony.

8We consider only a notion of identity that is based on reduction and normalisation. For furtheroptions, see Došen [8].


3.4 Towards a Definition of Strong Harmony

For simplicity, take the notion of reductive harmony mentioned in Sect. 3.2. Givenintroduction rules cI and elimination rules cE of an operator c, it associates withc the introduction meaning cI and the elimination meaning cE , and identifies exten-sional harmony with the equivalence of cI and cE in IPC2. It then appears to benatural to define intensional harmony as the availability of an isomorphism betweencI and cE in IPC2. However, this definition turns out to be unsuccessful, as thefollowing observation shows.

What we would like to achieve, in any case, is that the canonical eliminations forgiven introductions are in strong harmony with the introductions, and similarly thatthe canonical introductions for given eliminations are in strong harmony with theeliminations. The second case is trivial, as the premisses of the canonical introductionare exactly the conclusions of the eliminations. For example, if for the connective↔ the elimination rules

p1 ↔ p2 p1(↔ E)can p2

p1 ↔ p2 p2p1

are assumed to be given, then its canonical introduction rule has the form

[p1]p2

[p2]p1

(↔ I)can p1 ↔ p2

Both the elimination meaning ↔I and the introduction meaning ↔E have the form(p1 → p2) ∧ (p2 → p1), so that the identity proof identifies them. However, inthe first case, where the canonical eliminations are given by the general eliminationrules, the situation is more problematic.

Consider disjunction with the rules

p1p1 ∨ p2

p2p1 ∨ p2

p1 ∨ p2[p1]

q[p2]

qq

Here the introduction meaning of p1 ∨ p2 is p1 ∨ p2, viewed as a formula of IPC2,and its elimination meaning is ∀q(((p1 → q) ∧ (p2 → q)) → q). However,though p1 ∨ p2 and ∀q(((p1 → q) ∧ (p2 → q)) → q) are equivalent in IPC2,they are not isomorphic. There are proofs

p1 ∨ p2D1

∀q(((p1 → q) ∧ (p2 → q)) → q)

∀q(((p1 → q) ∧ (p2 → q)) → q)

D2p1 ∨ p2

of ∀q(((p1 → q) ∧ (p2 → q)) → q) from p1 ∨ p2 and of p1 ∨ p2 from∀q(((p1 → q) ∧ (p2 → q)) → q), so that the composition D2 ◦ D1 yields


the identity proof p1 ∨ p2, but there are no such proofs so that D1 ◦ D2 yieldsthe identity proof ∀q(((p1 → q) ∧ (p2 → q)) → q). One might object that,due to the definability of connectives in IPC2, p1 ∨ p2 should be understood as∀q(((p1 → q) ∧ (p2 → q)) → q), so that the isomorphism between p1∨ p2 and∀q(((p1 → q) ∧ (p2 → q)) → q) becomes trivial (and similarly, if conjunctionis also eliminated due to its definability in IPC2). To accommodate this objection, weconsider the example of the trivial connective+with the introduction and eliminationrules

p

+p+p

[p]q

q

Here the elimination rule is the canonical one according to the general-eliminationschema. In order to demonstrate strong harmony, we would have to establish theisomorphism of p and ∀q((p → q) → q) in IPC2, but this fails. This failure maybe related to the fact that for the second-order translations of propositional formulas,we do not have η-conversions in IPC2 (see Girard et al. [27], p. 859). This shows thatfor the definition of strong harmony the definition of introduction and eliminationmeaning by translation into IPC2 is perhaps not the best device. We consider thelack of an appropriate definition of strong harmony a major open problem, and weprovide two tentative solutions (with the emphasis on ‘tentative’).

First proposal: Complementation by canonical rules. In order to avoid theproblems of second-order logic, we can stay in intuitionistic propositional logic asfollows. Suppose for a constant c certain introduction rules cI and certain elimi-nation rules cE are proposed, and we ask: When are cI and cE in harmony witheach other? Suppose cI is the canonical elimination rule for the introduction rulescI , and cE is the canonical introduction rule for the elimination rules cE . We alsocall cI the canonical complement of cI , and cE the canonical complement of cE .We define two new connectives c1 and c2. Connective c1 has cI as its introductionrules and its complement cI as its elimination rule. Conversely, connective c2 hascE as its elimination rules and its complement cE as its introduction rules. In otherwords, for one connective we take the given introduction rules as complemented bythe canonical elimination rule, and for the other connective we take the given elim-ination rules as complemented by the canonical introduction rule. Furthermore, weassociate with c1 and c2 reduction procedures in the usual way, based on the pairscI /cI and cE /cE as primitive rules. Then we say that cI and cE are in strongharmony, if c1 is isomorphic to c2, that is, if there are proofs from c1 to c2 and back,such that the composition of these proofs is identical to the identity proof c1 or c2,depending on which side one starts with.10 In this way, by splitting up c into twoconnectives, we avoid the explicit translation into IPC2.11

9This was pointed out to me by Kosta Došen.10For better readability, we omit possible arguments of c1 and c2.11This procedure also works for weak harmony as a device to avoid the translation into second-orderlogic.


Second proposal: Change to the notion of canonical elimination.Asmentionedabove, we do not encounter any problem in IPC2, if we translate the introductionmeaning of disjunction by its disjunction-free second-order translation, as isomor-phism is trivial in this case. In fact, whenever we have more than one introductionrule for some c then the disjunction-free second-order translation is identical to thesecond-order translation of the eliminationmeaning for the canonical (indirect) elim-ination.Wehave a problem in the case of the connective+, which has the introductionmeaning p and the elimination meaning ∀q((p → q) → q). However, for + analternative elimination rule is derivable, namely the rule

+p

p

In fact, this sort of elimination rule is available for all connectives with only a singleintroduction rule. We call it the ‘direct’ as opposed to the ‘indirect’ elimination rule.For example, the connective &⊃ with the introduction rule

p1[p1]p2

p1 &⊃ p2

has as its direct elimination rules

p1 &⊃ p2p1

p1 &⊃ p2 p1p2

If we require that the canonical elimination rules always be direct where possible,that is, whenever there is not more than one introduction rule, and indirect only ifthere are multiple introduction rules, then the problem of reduction to IPC2 seems todisappear. In the direct case of a single introduction rule, the elimination meaning istrivially identical to the introduction meaning. In the indirect case, they now becometrivially identical again. This is because disjunction, which is used to express theintroduction meaning for multiple introduction rules, is translated into disjunction-free second-order logic in a way that makes its introduction meaning identical to itselimination meaning.

This second proposal would require the revision of basic tenets of proof-theoreticsemantics, because ever since the work of von Kutschera [36], Prawitz [47] andSchroeder-Heister [53] on general constants, and since the work on general elimina-tion rules, especially for implication, by Tennant [69, 70], López-Escobar [37] andvon Plato [43],12 the idea of the indirect elimination rules as the basic form of elimi-nation rules for all constants has been considered a great achievement. That said, theabandonment of projection-based conjunction and modus-ponens-based implicationhas received some criticism (Dyckhoff [15], Schroeder-Heister [66], Sect. 15.8). Infact, even the first proposal above might require this priority of the direct elimination

12For a discussion see Schroeder-Heister [65].


rules. If we consider conjunction with ∧I and ∧E given by the standard rules

p1 p2p1 ∧ p2

p1 ∧ p2p1

p1 ∧ p2p2

then ∧I is the generalised ∧-elimination rule

p1 ∧ p2[p1, p2]

qq

whereas ∧E is identical with ∧I . This means that strong harmony would requirethat projection-based conjunction and conjunction with general elimination are iso-morphic, but no such isomorphism obtains.13

4 Proof-Theoretic Semantics Beyond Logic

Proof-theoretic semantics has been occupied almost exclusively with logical reason-ing, and, in particular, with the meaning of logical constants. Even though the waywe can acquire knowledge logically is extremely interesting, this is not and shouldnot form the central pre-occupation of proof-theoretic semantics. The methods usedin proof-theoretic semantics extend beyond logic, often so that their application inlogic is nothing but a special case of these more general methods.

What is most interesting is the handling of reasoning with information that isincorporated into sentences, which, from the viewpoint of logic, are called ‘atomic’.A special way of providing such information, as long as we are not yet talking aboutempirical knowledge, is by definitions. By defining terms, we introduce claims intoour reasoning system that hold in virtue of the definition. In mathematics the mostprominent example is inductive definitions. Now definitional reasoning itself obeyscertain principles that we find otherwise in proof-theoretic semantics. As an inductivedefinition can be viewed as a set of rules the heads of which contain the definien-dum (for example, an atomic formula containing a predicate to be defined), it isonly natural to consider inductive clauses as kinds of introduction rules, suggestinga straightforward extension of principles of proof-theoretic semantics to the atomiccase. A particular challenge here comes from logic programming, where we con-sider inductive definitions of a certain kind, called ‘definite-clause programs’, anduse them not only for descriptive, but also for computational purposes. In the contextof dealing with negation, we even have the idea of inverting clauses in a certain sense.Principles such as the ‘completion’ of logic programs or the ‘closed-world assump-tion’ (which logic programming borrowed from Artificial Intelligence research), are

13This last observation is due to Luca Tranchini.


strongly related to principles generating elimination rules from introduction rulesand, thus, to the idea of harmony between these rules.

4.1 Definitional Reflection

In what follows, we sketch the idea of definitional reflection, which employs the ideaof clausal definitions as a powerful paradigm to extend proof-theoretic semanticsbeyond the specific realm of logic. It is related to earlier approaches developed byLorenzen [38, 39] who based logic (and also arithmetic and analysis) on a generaltheory of admissible rules using a sophisticated inversion principle (he coined theterm ‘inversion principle’ and was the first to formulate it in a precise way). It isalso related to Martin-Löf’s [40] idea of iterated inductive definitions, which givesintroduction and elimination rules for inductively defined atomic sentences. More-over, it is inspired by ideas in logic programming, where programs can be read asinductive definitions and where, in the attempt to provide a satisfactory interpreta-tion of negation, ideas that correspond to the inversion of rules have been considered(see Denecker et al. [5], Hallnäs and Schroeder-Heister [31]). We take definitionalreflection as a specific example of how proof-theoretic semantics can be extendedbeyond logic, and we claim that such an extension is quite useful. Other extensionsbeyond logic are briefly mentioned at the end of this section.

A particular advantage that distinguishes definitional reflection from theapproaches of Lorenzen and Martin-Löf and makes it more similar to what hasbeen done in logic programming is the idea that the meaning assignment by meansof a clausal or inductive definition can be partial, which means in particular thatdefinitions need not be well-founded. In logic programming this has been commonfrom the very beginning. For example, clauses such as

p ⇐ ¬p

which defines p by its own negation, or related circular clauses have been standardexamples for decades in the discussion of normal logic programs and the treat-ment of negation (see, e.g. Gelfond and Lifschitz [24], Gelder et al. [23]). Withinmainstream proof-theoretic semantics, such circular definitions have only recentlygarnered attention, in particular within the discussion of paradoxes, mostly with-out awareness of logic programming semantics and developments there. The ideaof definitional reflection can be used to incorporate smoothly partial meaning andnon-wellfounded definitions. We consider definitional reflection as an example ofhow to move beyond logic and, with it, beyond the totality and well-foundednessassumptions of the proof-theoretic semantics of logic.

As definitional reflection is a local approach not based on the placeholder viewof assumptions, we formulate it in a sequent-style framework. A definition is a listof clauses. A clause has the form

a ⇐ B


where the head a is an atomic formula (‘atom’). In the simplest case, the body B is alist of atoms b1, . . . , bm , inwhich case a definition looks like a definite logic program.We often consider an extended case where B may also contain structural implica-tion14 ‘⇒’, and sometimes even structural universal implication, which essentiallyis handled by restricting substitution. Given a definition D, the list of clauses with ahead starting with the predicate P is called the definition of P . In the propositionalcase where atoms are just propositional letters, we speak of the definition of a havingthe form

Da

⎧⎪⎨

⎪⎩

a ⇐ B1...

a ⇐ Bn

However, it should be clear that the definition of P or of a is normally just a particularpart of a definition D, which contains clauses for other expressions as well. It shouldalso be clear that this definition D cannot always be split up into separate definitionsof its predicates or propositional letters. So ‘definition of a’ or ‘of P’ is a mode ofspeech. What is always meant is the list of clauses for a predicate or propositionalletter within a definition D.

Syntactically, a clause resembles an introduction rule. However, in the theory ofdefinitional reflection we separate the definition, which is incorporated in the set ofclauses, from the inference rules, which put it into practice. So, instead of differentintroduction rules which define different expressions, we have a general schema thatapplies to a given definition. Separating the specific definition from the inferenceschema using arbitrary definitions gives us wider flexibility. We need not considerintroduction rules to be basic and other rules to be derived from them. Instead wecan speak of certain inference principles that determine the inferential meaning of aclausal definition andwhich are of equal stance. There is a pair of inference principlesthat put a definition into action, which are in harmony with each other, without oneof them being preferential. As we are working in a sequent-style framework, wehave inferential principles for introducing the defined constant on the right and onthe left of the turnstile, that is, in the assertion and in the assumption positions.For simplicity we consider the case of a propositional definition D, which has nopredicates, functions, individual variables or constants, and in which the bodies ofclauses are just lists of propositional letters. Suppose Da (as above) is the definition

14We speak of ‘structural’ implication to distinguish it from the implicational sentence connectivewhich may form part of a defined atom. Some remarks on this issue are made in Sect. 4.2.


of a (within D), and the Bi have the form ‘bi1, . . . , biki ’, as in propositional logicprogramming. Then the right-introduction rules for a are

� � bi1 . . . � � biki(�a)

� � a, in short

� � Bi

� � a(1 ≤ i ≤ n),

and the left-introduction rule for a is

�, B1 � C . . . �, Bn � C(a �)

�, a � C

If we talk generically about these rules, that is, without mentioning a specific a,but just the definition D, we also write (�D) and (D�). The right introduction ruleexpresses reasoning ‘along’ the clauses. It is also called definitional closure, bywhichis meant ‘closure under the definition’. The intuitive meaning of the left introductionrule is the following: Everything that follows from every possible definiens of a,follows from a itself. This rule is called the principle of definitional reflection, as itreflects upon the definition as a whole. If B1, . . . , Bn exhaust all possible conditionsto generate a according to the given definition, and if each of these conditions entailsthe very same conclusion, then a itself entails this conclusion.

This principle, which gives the whole approach its name, extracts deductive con-sequences of a from a definition in which only the defining conditions of a are given.If the clausal definition D is viewed as an inductive definition, definitional reflectioncan be viewed as being based on the extremal clause of D: Nothing else beyond theclauses given in D defines a. To give a very simple example, consider the followingdefinition: {

child_of_tom ⇐ annachild_of_tom ⇐ robert

Then one instance of the principle of definitional reflection with respect to thisdefinition is

anna� tall robert� tall

child_of_tom� tall

Therefore, if we know anna� tall and robert� tall, we can infer child_of_tom� tall.Since definitional reflection depends on the definition as a whole, taking all defini-

entia of a into account, it is non-monotonic with respect to D. If D is extended withan additional clause

a ⇐ Bn+1

for a, then previous applications of the (D�) rule may no longer remain valid. In thepresent example, if we add the clause

child_of_tom ⇐ john


we can no longer infer child_of_tom� tall, except when we also know john� tall.Note that due to the definitional reading of clauses, which gives rise to inversion,the sign ‘⇐’ expresses more than just implication, in contradistinction to structuralimplication ‘⇒’ that may occur in the body of a clause. To do justice to this fact,one might instead use ‘:-’ as in PROLOG, or ‘:=’ to express that we are dealing withsome sort of definitional equality.

In standard logic programming one has, on the declarative side, only what cor-responds to definitional closure. Definitional reflection leads to powerful extensionsof logic programming (due to computation procedures based on this principle) thatlie beyond the scope of the present discussion.

4.2 Logic, Paradoxes, Partial Definitions

Introduction rules (clauses) for logically compound formulas are not distinguishedin principle from introduction rules (clauses) for atoms. The introduction rules forconjunction and disjunction would, for example, be handled by means of clauses fora truth predicate with conjunction and disjunction as term-forming operators:

Dlog

⎧⎨

⎩

T (p ∧ q) ⇐ T (p), T (q)

T (p ∨ q) ⇐ T (p)

T (p ∨ q) ⇐ T (q)

In order to define implication, we need a rule arrow in the body, which, for the wholeclause, corresponds to using a higher-level rule:

T (p → q) ⇐ (T (p) ⇒ T (q))

This definition requires some sort of ‘background logic’. By that we mean the struc-tural logic governing the comma and the rule arrow ⇒, which determine how thebodies of clauses are handled. In standard logic we have just the comma, which ishandled implicitly. In extended versions of logic programming we would have the(iterated) rule arrow, that is, structural implication and associated principles govern-ing it, and perhaps even structural disjunction (this is present in disjunctive logicprogramming, but not needed for the applications considered here).

It is obvious that (�Dlog) gives us the right-introduction rules for conjunction anddisjunction or, more precisely, those for T (A), where A is a conjunction or disjunc-tion. Definitional reflection (Dlog �) gives us the left-introduction rules. The clausefor T (p → q) gives us the rules for implication, where the precise formulation ofthese rules depends on the exact formulation of the background logic governing ⇒.

Definitional reflection in general provides a much wider perspective on inversionprinciples than deductive logic alone. Using the definitional rule

t ∈ {x : a} ⇐ a[t/x]


we obtain a principle of naive comprehension, which does not lead to a useless theoryin which everything is derivable, even if we allow a to be the formula x /∈ x and tthe term {x : x /∈ x}. A definition of the form

p ⇐ ¬p

yields a paraconsistent system in which both � p and �¬p are derivable, withoutevery other formula being derivable. Formally, this means that the rule of cut

� � A A,�� B

�,�� B

is not always admissible. For special cases cut can be obtained, for example, if thedefinition is stratified, which essentially means that it is well-founded. So the well-behaviour of a definition in the case of logic, where we do have cut elimination, isdue to the fact that it obeys certain principles, which in the general case cannot beexpected to hold. This connects the proof theory of clausal definitionswith theories ofparadoxes, which conceive paradoxes as based on locally correct reasoning (Prawitz[44] (Appendix B), Tennant [68], Schroeder-Heister [62], Tranchini [71]).

For the situation that obtains here, Hallnäs [28] proposed the terms ‘total’ vs.‘partial’ in analogy with the terminology used in recursive function theory. That acomputable (i.e., partial recursive) function is total is not something required bydefinition, but is a matter of (mathematical) fact, actually an undecidable matter.Similarly, that a clausal definition yields a system that admits the elimination ofcuts is a result that may or may not hold true, but nothing that should enter therequirements for something to be admitted as a definition. If it holds, the definitionis called ‘total’, otherwise it is properly ‘partial’.

4.3 Variables and Substitution

The idea of proof-theoretic semantics beyond logic invites consideration of powerfulinversion principles that extend the simple form of definitional reflection consideredabove. Although wementioned clauses that contained variables, we formally defineddefinitional reflection only for propositional definitions of the form Da , where itsays that everything that can be inferred from each definiens can be obtained fromthe definiendum of a definition. However, this is insufficient for the more generalcase which is the standard case in logic programming. We show this by means ofan example. Suppose we have the following definition in which the atoms have apredicate-argument-structure:


⎧⎪⎪⎨

⎪⎪⎩

child_of_tom(anna) ⇐ daughter_of_tom(anna)child_of_tom(robert) ⇐ son_of_tom(robert)

tall(anna) ⇐ daughter_of_tom(anna)tall(robert) ⇐ son_of_tom(robert)

Given our propositional rule of definitional reflection, we could just infer proposi-tional results such as child_of_tom(anna)� tall(anna) or child_of_tom(robert)� tall(robert). However, what we would like to infer is the principle

child_of_tom(x)� tall(x)

with free variable x , since anna and robert are the only objects for which the predicatechild_of_tom is defined, and since for them the desired principle holds.

In an even more general case, we have clauses that contain variables the instancesof which match instances of the claim we want to obtain by definitional reflection.Consider the definition

D1

{p(y, a) ⇐ q(y)

p(x, f (a)) ⇐ q( f (x))

According to the principle of general definitional reflection, we obtain, with respectto D1,

p(a, z)� q(z)

The intuitive argument is as follows: Suppose p(a, z). Any, and in fact the only,substitution instance of p(a, z) that can be obtained by the first clause is generatedby substituting a for y in the first clause, and a for z in p(a, z). We denote thissubstitution by [a/y, a/z] and call itσ1.Any, and in fact the only, substitution instanceof p(a, z) that can be obtained by the second clause is generated by substituting afor x in the second clause, and f (a) for z in p(a, z). We denote this substitution by[a/x, f (a)/z] and call it σ2. When σ1 is applied to the body of the first clause, q(a)

is obtained, which is also obtained when σ1 is applied to q(z). When σ2 is appliedto the body of the second clause, q( f (a)) is obtained, which is also obtained whenσ2 is applied to q(z). Therefore, we can conclude q(z).

In the propositional case we could describe definitional reflection by saying thatevery C that is a consequence of each defining condition is a consequence of thedefiniendum a. We cannot now even identify a single formula as a definiendum, asany formula which is a substitution instance of a head of a definitional clause isconsidered to be defined. Therefore we should now say: Suppose a formula a isgiven. If for each substitution instance aσ that can be obtained as bσ from the headof a clause b ⇐ C , we have that Cσ implies Aσ for some A, then A can be inferredfrom a.


Formally, this leads to a principle of definitional reflection according to which,for the introduction of an atom a on the left side of the turnstile, the most generalunifiers mgu(a, b) of a with the heads of all definitional clauses are considered:

(D�)ω{�σ, Cσ � Aσ : σ = mgu(a, b) for some clause b ⇐ C in D}

�, a � A

with the proviso: The variables free in �, a � A must be different from those in theb ⇐ C above the line. This means that we always assume that variables in clausesare standardised apart. We call this principle the ω-version of definitional reflection,as it is in a certain way related to the ω-rule in arithmetic, a point that we cannotelaborate on here (see Schroeder-Heister [55]).

This powerful principle is typical of applications outside logic.When we considerlogical definitions such asDlog,we see that each formulaT (a),wherea is a compoundlogical formula, determines exactly a single head b in one or two clauses (two in thecase of disjunction), such that T (a) is a substitution instance of b. This means that(1) there is just matching and no unification between b and T (a), and (2) there isjust a single substitution for the main logical connective in T (a) involved due to thestrict separation between the clauses for the different logical connectives, implyingthat there is no overlap between substitution instances of clauses.

The power of theω-version of definitional reflection is demonstrated, for example,by the fact that the rules of free equality can be obtained from the definition consistingof the single clause

D= { x = x ⇐

For example, the transitivity of equality is derived by a single inference step asfollows:

x2 = x3 � x2 = x3(D= �)ω x1 = x2, x2 = x3 � x1 = x3

Here we use that the substitution [x2/x1, x2/x] is an mgu of x1 = x2 with the headx = x of the clause x = x ⇐ . In a similar way, all freeness axioms of Clark’sequational theory [35] can be derived.15

4.4 Outlook: Applications and Extensions of DefinitionalReflection

We have not discussed computational issues here. Clausal definitions give rise tocomputational procedures as investigated and implemented in logic programming,

15For further discussion see Girard [26], Schroeder-Heister [32], Eriksson [16, 17], Schroeder-Heister [54]. Of all inversion principles mentioned in the literature, only Lorenzen’s original one[39] comes close to the power of definitional reflection (though substantial differences remain, seeSchroeder-Heister [57]).


and definitional reflection adds a strong component to such computation (see Hallnäsand Schroeder-Heister [31], Eriksson [17]). This computational aspect is importantto proof-theoretic semantics. We should not only be able to give a definition of asemantically correct proof, where ‘semantically’ is understood in the sense of proof-theoretic semantics, we should also be interested inways to construct such proofs thatproceed according to such principles. Programming languages, theorem provers andproof editors based on inversion principles make important contributions to this task.Devising principles of proof construction that can be used for proof search is itselfan issue of proof-theoretic semantics that is a desideratum in the philosophicallydominated community. Theories that go beyond logic are of particular interest here,as theorems outside pure logic are what we normally strive for in reasoning.

If we want to deal with more advanced mathematical theories, stronger closureand reflection principles are needed. At an elementary level, clauses a ⇐ B in adefinition can be used to describe function computation in the form

f (x1, . . . , xk) ⇐ (x1, . . . , xk)

which is supposed to express that from the arguments x1, . . . , xk the value f (x1,. . . , xk) is obtained, so that by means of definitional reflection f (x1, . . . , xk) can becomputed. More generally, one might describe functionals F by means of (infini-tary) clauses the bodies of which describe the evaluation of functions f which arearguments of F (for some hints see Hallnäs [29, 30]). An instructive example is theanalysis of abstract syntax (see McDowell and Miller [41]).

There are several other approaches that deal with the atomic level proof-theo-retically, that is, with issues beyond logic in the narrower sense. These approachesinclude Negri and von Plato’s [42] proof analysis, Brotherston and Simpson’s[2] infinite derivations, or even derivations concerning subatomic expressions (seeWieckowski [73]), and corresponding linguistic applications, as discussed byFrancezand Dyckhoff [19] and Francez et al. [20]. Proof-theoretic semantics beyond logicis a broad field with great potential, the surface of which, thus far, has barely beenscratched.

Acknowledgments I am grateful to Kosta Došen, Thomas Piecha, Dag Prawitz, Luca Tranchiniand John William Devine for helpful comments and suggestions. The completion of this paper wassupported by the French-German ANR-DFG project ‘Beyond Logic: Hypothetical Reasoning inPhilosophy of Science, Informatics, and Law’ (DFG Schr 275/17-1).

Open Access This chapter is distributed under the terms of the Creative Commons AttributionNoncommercial License, which permits any noncommercial use, distribution, and reproduction inany medium, provided the original author(s) and source are credited.


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